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M O D E R N Q U A N T U M MECH A NICS S E C O N D E D I T I O N (cid:43)(cid:15)(cid:1)(cid:43)(cid:15)(cid:1)(cid:52)(cid:66)(cid:76)(cid:86)(cid:83)(cid:66)(cid:74)(cid:1)(cid:1)(cid:116)(cid:1)(cid:1)(cid:43)(cid:74)(cid:78)(cid:1)(cid:47)(cid:66)(cid:81)(cid:80)(cid:77)(cid:74)(cid:85)(cid:66)(cid:79)(cid:80) Modern Quantum Mechanics Second Edition ModernQuantumMechanicsisaclassicalgraduateleveltextbook,coveringthemainquantummechan- ics concepts in a clear, organized and engaging manner. The author, J. J. Sakurai, was a renowned theorist in particle theory. The Second Edition, revised by Jim Napolitano, introduces topics that extend the text’s usefulness into the 21st century such as advanced mathematical techniques associ- ated with quantum mechanical calculations, while at the same time retaining classic developments such as neutron interferometer experiments, Feynman path integrals, correlation measurements, and Bell’s inequality. A solution manual for instructors using this textbook can be downloaded from www.cambirdge.org/napolitanoundertheresourcestab. ThelateJ.J.Sakurai,notedtheoristinparticlephysics,wasborninTokyo,Japanin1933.Hereceived his B.A. from Harvard University in 1955 and his PhD from Cornell University in 1958. He was appointed as an assistant professor at the University of Chicago, where he worked until he became a professor at the University of California, Los Angeles in 1970. Sakurai died in 1982 while he was visitingaprofessoratCERNinGeneva,Switzerland. Jim Napolitano earned an undergraduate Physics degree at Rensselaer Polytechnic Institute in 1977, andaPhDinPhysicsfromStanfordUniversityin1982.Sincethattime,hehasconductedresearchin experimentalnuclearandparticlephysics,withanemphasisonstudyingfundamentalinteractionsand symmetries.HejoinedthefacultyatRensselaerin1992afterworkingasamemberofthescientificstaff attwodifferentnationallaboratories.Since2014hehasbeenProfessorofPhysicsatTempleUniversity. He is author and co-author of over 150 scientific papers in refereed journals. Professor Napolitano maintainsakeeninterestinscienceeducationingeneral,andinparticularphysicseducationatboththe undergraduateandgraduatelevels.Hehastaughtbothgraduateandupper-levelundergraduatecourses inQuantumMechanics,aswellasanadvancedgraduatecourseinQuantumFieldTheory. Modern Quantum Mechanics Second Edition J. J. Sakurai Deceased Jim Napolitano TempleUniversity,Philadelphia UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom OneLibertyPlaza,20thFloor,NewYork,NY10006,USA 477WilliamstownRoad,PortMelbourne,VIC3207,Australia 4843/24,2ndFloor,AnsariRoad,Daryaganj,Delhi–110002,India 79AnsonRoad,#06–04/06,Singapore079906 CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learning,andresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781108422413 DOI:10.1017/9781108499996 (cid:2)c CambridgeUniversityPress2017 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. ThisbookwaspreviouslypublishedbyPearsonEducation,Inc.1994,2011 ReissuedbyCambridgeUniversityPress2017 PrintedintheUnitedKingdombyTJInternationalLtd.PadstowCornwall AcataloguerecordforthispublicationisavailablefromtheBritishLibrary. Additionalresourcesforthispublicationavailableat:www.cambridge.org/napolitano ISBN978-1-108-42241-3Hardback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain, accurateorappropriate. Contents Foreword to the First Edition ix Preface to the Revised Edition xi Preface to the Second Edition xiii In Memoriam xvii 1 Fundamental Concepts 1 1.1 TheStern-GerlachExperiment 1 1.2 Kets,Bras,andOperators 10 1.3 BaseKetsandMatrixRepresentations 17 1.4 Measurements,Observables,andtheUncertaintyRelations 23 1.5 ChangeofBasis 35 1.6 Position,Momentum,andTranslation 40 1.7 WaveFunctionsinPositionandMomentumSpace 50 2 Quantum Dynamics 66 2.1 Time-EvolutionandtheSchrödingerEquation 66 2.2 TheSchrödingerVersustheHeisenbergPicture 80 2.3 SimpleHarmonicOscillator 89 2.4 Schrödinger’sWaveEquation 97 2.5 ElementarySolutionstoSchrödinger’sWaveEquation 103 2.6 PropagatorsandFeynmanPathIntegrals 116 2.7 PotentialsandGaugeTransformations 129 3 Theory of Angular Momentum 157 3.1 RotationsandAngular-MomentumCommutationRelations 157 3.2 Spin 1 SystemsandFiniteRotations 163 2 3.3 SO(3),SU(2),andEulerRotations 172 v vi Contents 3.4 DensityOperatorsandPureVersusMixedEnsembles 178 3.5 EigenvaluesandEigenstatesofAngularMomentum 191 3.6 OrbitalAngularMomentum 199 3.7 Schrödinger’sEquationforCentralPotentials 207 3.8 AdditionofAngularMomenta 217 3.9 Schwinger’sOscillatorModelofAngularMomentum 232 3.10 SpinCorrelationMeasurementsandBell’sInequality 238 3.11 TensorOperators 246 4 Symmetry in Quantum Mechanics 262 4.1 Symmetries,ConservationLaws,andDegeneracies 262 4.2 DiscreteSymmetries,Parity,orSpaceInversion 269 4.3 LatticeTranslationasaDiscreteSymmetry 280 4.4 TheTime-ReversalDiscreteSymmetry 284 5 Approximation Methods 303 5.1 Time-IndependentPerturbationTheory:NondegenerateCase 303 5.2 Time-IndependentPerturbationTheory:TheDegenerateCase 316 5.3 Hydrogen-LikeAtoms:FineStructureandtheZeemanEffect 321 5.4 VariationalMethods 332 5.5 Time-DependentPotentials:TheInteractionPicture 336 5.6 HamiltonianswithExtremeTimeDependence 345 5.7 Time-DependentPerturbationTheory 355 5.8 ApplicationstoInteractionswiththeClassicalRadiationField 365 5.9 EnergyShiftandDecayWidth 371 6 Scattering Theory 386 6.1 ScatteringasaTime-DependentPerturbation 386 6.2 TheScatteringAmplitude 391 6.3 TheBornApproximation 399 6.4 PhaseShiftsandPartialWaves 404 6.5 EikonalApproximation 417 6.6 Low-EnergyScatteringandBoundStates 423 6.7 ResonanceScattering 430 6.8 SymmetryConsiderationsinScattering 433 6.9 InelasticElectron-AtomScattering 436 7 Identical Particles 446 7.1 PermutationSymmetry 446 7.2 SymmetrizationPostulate 450 Contents vii 7.3 Two-ElectronSystem 452 7.4 TheHeliumAtom 455 7.5 MultiparticleStates 459 7.6 QuantizationoftheElectromagneticField 472 8 Relativistic Quantum Mechanics 486 8.1 PathstoRelativisticQuantumMechanics 486 8.2 TheDiracEquation 494 8.3 SymmetriesoftheDiracEquation 501 8.4 SolvingwithaCentralPotential 506 8.5 RelativisticQuantumFieldTheory 514 A Electromagnetic Units 519 A.1 Coulomb’sLaw,Charge,andCurrent 519 A.2 ConvertingBetweenSystems 520 B Brief Summary of Elementary Solutions to Schrödinger’s Wave Equation 523 B.1 FreeParticles(V =0) 523 B.2 PiecewiseConstantPotentialsinOneDimension 524 B.3 Transmission-ReflectionProblems 525 B.4 SimpleHarmonicOscillator 526 B.5 TheCentralForceProblem[SphericallySymmetricalPotential V =V(r)] 527 B.6 HydrogenAtom 531 C Proof of the Angular-Momentum Addition Rule Given by Equation (3.8.38) 533 Bibliography 535 Index 537